Prime Filter is Prime Ideal in Dual Lattice

Theorem

Let $L = \struct {S, \preceq}$ be a lattice.

Let $X$ be a subset of $S$.

Then

$X$ is a prime filter in $L$

if and only if:

$X$ is a prime ideal in $L^{-1}$

where $L^{-1} = \struct {S, \succeq}$ denotes the dual of $L$.

Proof

By Dual of Dual Ordering:

dual of $L^{-1}$ is $L$.

Hence by Prime Ideal is Prime Filter in Dual Lattice:

the result follows.

$\blacksquare$

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL_7:17